TRANSFORMATIONS

DEFINITION  A TRANSFORMATION is a change in a figure’s position or size.  An Image is the resulting figure of a translation, rotation, or reflection.

3 TYPES OF TRANSFORMATIONS  1. Translations  2. Rotations  3. Reflections

TRANSLATION  A translation slides a figure along a line without turning.  Key Word: SLIDES

TRANSLATION RULES TypeRule Move right a unitsAdd a to each x- coordinate ( x, y)  ( x + a, y) Move left a unitsSubtract a from each x- coordinate: ( x,y)  (x – a, y) Move up b unitsAdd b to each y- coordinate (x, y)  (x, y + b) Move down b units Subtract b from each y- coordinate ( x, y)  (x, y – b)

EXAMPLE # 1  Graph the translation of ∆ ABC 3 units left and 4 units up. Subtract 3 from the x- coordinate of each vortex, and add 4 to the y- coordinate of each vertex. RuleImage A(1, -1)  A’ (1 – 3, )A’( -2, 3) B(1,- 3)  B’(1 – 3, )B’ (-2, 1) C (4, -3)  C’ (4 – 3, )C’ (1, 1)

EXAMPLE #2  Graph the translation 2 units right and 3 units up

EXAMPLE # 3  4 units right and 1 unit down

BELL WORK  Draw the image of a triangle with vertices (-1, 2), (3, 3), and (1, -3) after a translation 2 units up and 3 units right.

REFLECTIONS  A Reflection flips a figure across a line to create a mirror image.  In this example the triangle is reflected across the Y- axis. Notice that the x- coordinates of corresponding Vertices are opposite and that the y- coordinates stay The same. This suggests a rule that can be used to Reflect figures across either axis.

REFLECTION RULES TypeRule Across the y- axis Multiply each x- coordinate by -1: (x, y)  (-x, y) Across the x- axis Multiply each y- coordinate by -1: (x, y)  (x, -y)

EXAMPLE #1  Graph the reflection of quadrilaterals ABCD across the x- axis.

DILATIONS  A dilation is a transformation that changes the size, but not the shape, of a figure.  After a dilation, the image is similar to the original figure.

CENTER OF DILATION  The center of dilation is the fixed point of every dilation.  To find the center of dilation, draw lines that connect each pair of corresponding vertices. The lines intersect at one point that is called the center of dilation.

TELL WHETHER EACH TRANSFORMATION IS A DILATION. EXPLAIN. Compare the ratios of corresponding side lengths. Yes; the ratios of corresponding side length are equal and corresponding angles are congruent.

TELL WHETHER EACH TRANSFORMATION IS A DILATION. EXPLAIN. Compare the ratios of corresponding side lengths

SCALE FACTOR  The scale factor describes how much a figure is enlarged or reduced.  It represents the ratio of a length on the image to the corresponding length on the original figure

RULE FOR DILATION  For a dilation centered at the origin with scale factor k, the image of point P(x,y) is found by multiplying each coordinate by k.  ( x, y)  ( kx, ky)  If k > 1, then the image is larger than the preimage.  If 0 < k < 1, then the image is smaller than the preimage.

Dilate the figure by a scale factor of 2. What are the vertices of the image? First thing: Plot each coordinate point: A: ( 2, 2) B: ( 3, 4) C: ( 5, 2) Next: Multiply the coordinates by 2 to find the vertices of the image. A’: B’: C’: